Bad Solenoid?

I would first remove my battery connections at both ends pos. and neg. Inspect, clean both surfaces and reinstall. If it still clicks I would then charge the battery for couple hours also check the water level in the battery if it is accessable). If it still clicks I would remove the battery take it to autozone or similar auto parts store and have the battery load tested. If it is full charged and load tests good and still clicks after you reinstall it, now I would think about replacing some parts after I jumped around the solenoid to see if it starts or not.

I understand and agree. Should I then test for voltage at the battery to switch, then switch to solenoid and then solenoid to starter?
If I hear the click with switch on and the starter does not engage with the switch on start and the starter does not engage but will engage when jumped...solenoid bad?

***"but will engage when jumped...solenoid bad?"***
That depends on what you call "Jump".
If you are jumping across both large posts of the solenoid you are essentially only testing the starter motor.
If you jumper from the battery cable post on the starter solenoid to the small wire post on the starter solenoid and the solenoid fails to click, there is a good chance the solenoid is defective.
If the solenoid clicks and operates the starter when you jumper the same way..........but will not click when you turn the key to start, you may have a defective key switch or brake switch.
What kept you from posting the brand and model number of the tractor?
With the right info from you we might have a chance of finding a wiring diagram for it and then maybe telling you exactly how to test things.
If you have not already bought a bunch of parts, you might still have a chance to keep at least some of your money.

In Bill's test, the solenoid is mounted in its normal place on the subject machine. All electrical connections (wires/cables) are exactly as they are supposed to be.
Connect the grounding lead of a test light to one large post of the solenoid and the test light probe to the other large post of the solenoid. If the starter internal wiring is good, the test light will light up because it is using the starter internal wiring to reach the ground side of the battery.
Then........when the solenoid is operated electrically (by key switch or jumper wire activation) and it "clicks", the light will go out if the solenoid is good because the large contacts inside the solenoid offer an easier (less resistance) path for the current to follow instead of traveling through the test light as before.
The light first burned because it offered a path from the battery + post of the solenoid......... to the battery - post ground. If the solenoid clicks....and is good..... the current will abandon its previous path of high resistance through the light and take the easy path through the internal circuit of the solenoid....and the light will stop burning.

They are fairly "generic" in that if the size of the big studs is the same, and the number and configuration of any small studs are identical............ the solenoid will work.
Where you might have problems is if your replacement solenoid has 2 small posts instead of just 1.
The "work around" trick for that is to use a short "grounding wire" to connect the second small post to a good grounding bolt or screw that fastens into the frame or engine.
You might also have to accommodate a different mounting footprint by drilling a different hole for a mounting screw if the base does not match the OEM solenoid. Possibilities exist.
Are you doing any of the suggested tests or are you gonna try a spare solenoid? Inquiring minds want to know.